Many students I have spoken with who are drawn to becoming mathematics teachers chose their mathematics major because they enjoyed doing routine exercises in high school. The comfort of a definite and systematic general procedure that eliminates uncertainty is somehow a major motivation for these students to pursue mathematics.

The above motivation stands in stark contrast to the attitudes of math olympiad enthusiasts and Eastern European colleagues raised on a diet of competition-type problems. It seems to me that the major motivation for these sorts of mathematics students is to collect an enormous store of problems that each stand out because of a distinguishing solution twist.

Being overly simple minded, let me assume that for the first group the comfort in mathematics comes from the general coherence of a system, and from the second it comes from the charm of particular phenomena. (Forgive the binary nature of this, but I would like to use it as a way to tease out what I want to isolate.)

Q: Are there papers in the literature that study the motivation of students for studying mathematics in terms of their placement on the "general-particular" spectrum suggested by the above extremes? If so, can someone please point me to some of these papers?

I ask this because I'm interested in "converting" some students from the first to second motivating viewpoint, and the first point of view seems rather persistent. (I am trying to do this not because I believe that the problem solving culture is necessarily "best", but that it seems a better alternative to the first point of view.)

Edit: On a somewhat related note that I will post here because it should not take up space elsewhere, at the root of this question is the more fundamental question of "sustainable mathematical motivation". Making the (certainly misguided) assumption that the "telos" of a mathematics student should be a productive research mathematician, the "general" or perhaps "scientific" viewpoint may be more sustainable (think about a Bourbaki approach) since one can follow naturality and pretty much continuously record observations about a mathematical question until something comes out of it…and this can probably be done steadily for an entire career, assuming one does not become disenchanted facing the tremendous stamina needed for such an approach. On the other hand, the ability to find delight in shorter bursts throughout the problemist literature can lead to the continual building of technical strength that can be brought to bear on many different problems, and such an approach is perhaps more sustainable due to the little bursts of drama found in each problem. In reality, mathematicians lie on a spectrum in their tastes and time-management inclinations…so the present question is slanted by the assumption that the latter sort of taste is the more human, and thus more sustainable. (A bad Erdos is better than a bad Grothendieck...)

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    $\begingroup$ I would suggest a third way: collecting narratives. I certainly do not care about memorizing systematic solutions to particular general sorts of problems. I also do not care much for "tricky" solutions needed for particular problems. I much prefer developing stories: This kind of problem seems hard because of X, but when you think of it in context Y, we can now apply method Z, and so everything becomes clear. $\endgroup$ – Steven Gubkin Feb 17 '16 at 22:13
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    $\begingroup$ It seems to me that the first point of view is more scientific (reducing seemingly disparate phenomena to orderly patterns), whereas the second point of view is more narrative (tell a familiar story with a twist). I would argue that the first point of view is more powerful, as demonstrated by Kepler and Mendeleev. In practice, where along this spectrum are people most likely to test hypotheses? $\endgroup$ – Jasper Feb 18 '16 at 0:16
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    $\begingroup$ The two groups described in the problem don't sound like anyone I've ever met. This applies especially to the first group. Are there really people who enjoy solving routine exercises using predefined algorithms? That's what computers are for. If there are such people, I would imagine that they weren't very bright, or maybe that they were on the autism spectrum. $\endgroup$ – Ben Crowell Feb 18 '16 at 0:51
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    $\begingroup$ @BenCrowell I regularly meet people who say that they enjoy math for this reason. $\endgroup$ – Steven Gubkin Feb 18 '16 at 2:03
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    $\begingroup$ @BenCrowell "Are there really people who enjoy solving routine exercises using predefined algorithms?" Just going to pile on here: Oh god yes. I think it's an open theory that perhaps most people who want to be secondary-school math teachers may be in that category. $\endgroup$ – Daniel R. Collins Aug 11 '16 at 0:07

To the question about whether or not there is existant literature around some of these ideas, I believe there is a great deal. It seems you are getting at conversations around proof and how this plays out in teachers beliefs about mathematics. Several researchers have engaged in such pursuits. Immediately two things come to mind.

  • $\begingroup$ Could you perhaps edit your answer to provide more detail, such as a brief summary of the research, along with a clearer explanation of how this fits in with the "general-particular" spectrum mentioned by the OP? $\endgroup$ – J W Aug 17 '16 at 13:10

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